A lot of times proving your case involves asking your children to testify against one of their parents.
That is a bridge too far for a lot of people.
That's baloney. An annulment proceeding is an examination of the conditions present at the time of the wedding. What happened afterward is irrelevant to the question of whether the marriage itself was valid. There is no reason that anyone would need to "testify against" someone else. They only thing "on trial" is the validity of the marriage itself. Assuming that the children were born after the wedding, nothing they would personally be able to testify about would be relevant.
The Church brings no harm to innocents in the case of an investigation and is strictly business, free of accusation, innuendo and high drama. Mostly the Tribunal must verify dates and times and addresses and contacts witness who can confirm the basic information given them by the petitioner as relates to records.
The petioner recommends witnesses, and also the Tribunal accesses church records for support of the petitioner. Who, what, where, when, and why and how sorts of questions.
Someone from the Tribunal is also appointed to argue against granting the annulment, so that both sides of a petitioners case may be argued in the face of the evidence and the narrow criteria for granting annulments for a full and fair hearing.